// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_REAL_SCHUR_H
#define EIGEN_REAL_SCHUR_H

#include "./HessenbergDecomposition.h"

namespace Eigen {

/** \eigenvalues_module \ingroup Eigenvalues_Module
 *
 *
 * \class RealSchur
 *
 * \brief Performs a real Schur decomposition of a square matrix
 *
 * \tparam _MatrixType the type of the matrix of which we are computing the
 * real Schur decomposition; this is expected to be an instantiation of the
 * Matrix class template.
 *
 * Given a real square matrix A, this class computes the real Schur
 * decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
 * T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
 * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
 * matrix is a block-triangular matrix whose diagonal consists of 1-by-1
 * blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
 * blocks on the diagonal of T are the same as the eigenvalues of the matrix
 * A, and thus the real Schur decomposition is used in EigenSolver to compute
 * the eigendecomposition of a matrix.
 *
 * Call the function compute() to compute the real Schur decomposition of a
 * given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
 * constructor which computes the real Schur decomposition at construction
 * time. Once the decomposition is computed, you can use the matrixU() and
 * matrixT() functions to retrieve the matrices U and T in the decomposition.
 *
 * The documentation of RealSchur(const MatrixType&, bool) contains an example
 * of the typical use of this class.
 *
 * \note The implementation is adapted from
 * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
 * Their code is based on EISPACK.
 *
 * \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver
 */
template<typename _MatrixType>
class RealSchur
{
  public:
	typedef _MatrixType MatrixType;
	enum
	{
		RowsAtCompileTime = MatrixType::RowsAtCompileTime,
		ColsAtCompileTime = MatrixType::ColsAtCompileTime,
		Options = MatrixType::Options,
		MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
		MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
	};
	typedef typename MatrixType::Scalar Scalar;
	typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
	typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3

	typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
	typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;

	/** \brief Default constructor.
	 *
	 * \param [in] size  Positive integer, size of the matrix whose Schur decomposition will be computed.
	 *
	 * The default constructor is useful in cases in which the user intends to
	 * perform decompositions via compute().  The \p size parameter is only
	 * used as a hint. It is not an error to give a wrong \p size, but it may
	 * impair performance.
	 *
	 * \sa compute() for an example.
	 */
	explicit RealSchur(Index size = RowsAtCompileTime == Dynamic ? 1 : RowsAtCompileTime)
		: m_matT(size, size)
		, m_matU(size, size)
		, m_workspaceVector(size)
		, m_hess(size)
		, m_isInitialized(false)
		, m_matUisUptodate(false)
		, m_maxIters(-1)
	{
	}

	/** \brief Constructor; computes real Schur decomposition of given matrix.
	 *
	 * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
	 * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
	 *
	 * This constructor calls compute() to compute the Schur decomposition.
	 *
	 * Example: \include RealSchur_RealSchur_MatrixType.cpp
	 * Output: \verbinclude RealSchur_RealSchur_MatrixType.out
	 */
	template<typename InputType>
	explicit RealSchur(const EigenBase<InputType>& matrix, bool computeU = true)
		: m_matT(matrix.rows(), matrix.cols())
		, m_matU(matrix.rows(), matrix.cols())
		, m_workspaceVector(matrix.rows())
		, m_hess(matrix.rows())
		, m_isInitialized(false)
		, m_matUisUptodate(false)
		, m_maxIters(-1)
	{
		compute(matrix.derived(), computeU);
	}

	/** \brief Returns the orthogonal matrix in the Schur decomposition.
	 *
	 * \returns A const reference to the matrix U.
	 *
	 * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
	 * member function compute(const MatrixType&, bool) has been called before
	 * to compute the Schur decomposition of a matrix, and \p computeU was set
	 * to true (the default value).
	 *
	 * \sa RealSchur(const MatrixType&, bool) for an example
	 */
	const MatrixType& matrixU() const
	{
		eigen_assert(m_isInitialized && "RealSchur is not initialized.");
		eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
		return m_matU;
	}

	/** \brief Returns the quasi-triangular matrix in the Schur decomposition.
	 *
	 * \returns A const reference to the matrix T.
	 *
	 * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
	 * member function compute(const MatrixType&, bool) has been called before
	 * to compute the Schur decomposition of a matrix.
	 *
	 * \sa RealSchur(const MatrixType&, bool) for an example
	 */
	const MatrixType& matrixT() const
	{
		eigen_assert(m_isInitialized && "RealSchur is not initialized.");
		return m_matT;
	}

	/** \brief Computes Schur decomposition of given matrix.
	 *
	 * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
	 * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
	 * \returns    Reference to \c *this
	 *
	 * The Schur decomposition is computed by first reducing the matrix to
	 * Hessenberg form using the class HessenbergDecomposition. The Hessenberg
	 * matrix is then reduced to triangular form by performing Francis QR
	 * iterations with implicit double shift. The cost of computing the Schur
	 * decomposition depends on the number of iterations; as a rough guide, it
	 * may be taken to be \f$25n^3\f$ flops if \a computeU is true and
	 * \f$10n^3\f$ flops if \a computeU is false.
	 *
	 * Example: \include RealSchur_compute.cpp
	 * Output: \verbinclude RealSchur_compute.out
	 *
	 * \sa compute(const MatrixType&, bool, Index)
	 */
	template<typename InputType>
	RealSchur& compute(const EigenBase<InputType>& matrix, bool computeU = true);

	/** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T
	 *  \param[in] matrixH Matrix in Hessenberg form H
	 *  \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
	 *  \param computeU Computes the matriX U of the Schur vectors
	 * \return Reference to \c *this
	 *
	 *  This routine assumes that the matrix is already reduced in Hessenberg form matrixH
	 *  using either the class HessenbergDecomposition or another mean.
	 *  It computes the upper quasi-triangular matrix T of the Schur decomposition of H
	 *  When computeU is true, this routine computes the matrix U such that
	 *  A = U T U^T =  (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
	 *
	 * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
	 * is not available, the user should give an identity matrix (Q.setIdentity())
	 *
	 * \sa compute(const MatrixType&, bool)
	 */
	template<typename HessMatrixType, typename OrthMatrixType>
	RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU);
	/** \brief Reports whether previous computation was successful.
	 *
	 * \returns \c Success if computation was successful, \c NoConvergence otherwise.
	 */
	ComputationInfo info() const
	{
		eigen_assert(m_isInitialized && "RealSchur is not initialized.");
		return m_info;
	}

	/** \brief Sets the maximum number of iterations allowed.
	 *
	 * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
	 * of the matrix.
	 */
	RealSchur& setMaxIterations(Index maxIters)
	{
		m_maxIters = maxIters;
		return *this;
	}

	/** \brief Returns the maximum number of iterations. */
	Index getMaxIterations() { return m_maxIters; }

	/** \brief Maximum number of iterations per row.
	 *
	 * If not otherwise specified, the maximum number of iterations is this number times the size of the
	 * matrix. It is currently set to 40.
	 */
	static const int m_maxIterationsPerRow = 40;

  private:
	MatrixType m_matT;
	MatrixType m_matU;
	ColumnVectorType m_workspaceVector;
	HessenbergDecomposition<MatrixType> m_hess;
	ComputationInfo m_info;
	bool m_isInitialized;
	bool m_matUisUptodate;
	Index m_maxIters;

	typedef Matrix<Scalar, 3, 1> Vector3s;

	Scalar computeNormOfT();
	Index findSmallSubdiagEntry(Index iu, const Scalar& considerAsZero);
	void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift);
	void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
	void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
	void performFrancisQRStep(Index il,
							  Index im,
							  Index iu,
							  bool computeU,
							  const Vector3s& firstHouseholderVector,
							  Scalar* workspace);
};

template<typename MatrixType>
template<typename InputType>
RealSchur<MatrixType>&
RealSchur<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeU)
{
	const Scalar considerAsZero = (std::numeric_limits<Scalar>::min)();

	eigen_assert(matrix.cols() == matrix.rows());
	Index maxIters = m_maxIters;
	if (maxIters == -1)
		maxIters = m_maxIterationsPerRow * matrix.rows();

	Scalar scale = matrix.derived().cwiseAbs().maxCoeff();
	if (scale < considerAsZero) {
		m_matT.setZero(matrix.rows(), matrix.cols());
		if (computeU)
			m_matU.setIdentity(matrix.rows(), matrix.cols());
		m_info = Success;
		m_isInitialized = true;
		m_matUisUptodate = computeU;
		return *this;
	}

	// Step 1. Reduce to Hessenberg form
	m_hess.compute(matrix.derived() / scale);

	// Step 2. Reduce to real Schur form
	// Note: we copy m_hess.matrixQ() into m_matU here and not in computeFromHessenberg
	//       to be able to pass our working-space buffer for the Householder to Dense evaluation.
	m_workspaceVector.resize(matrix.cols());
	if (computeU)
		m_hess.matrixQ().evalTo(m_matU, m_workspaceVector);
	computeFromHessenberg(m_hess.matrixH(), m_matU, computeU);

	m_matT *= scale;

	return *this;
}
template<typename MatrixType>
template<typename HessMatrixType, typename OrthMatrixType>
RealSchur<MatrixType>&
RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH,
											 const OrthMatrixType& matrixQ,
											 bool computeU)
{
	using std::abs;

	m_matT = matrixH;
	m_workspaceVector.resize(m_matT.cols());
	if (computeU && !internal::is_same_dense(m_matU, matrixQ))
		m_matU = matrixQ;

	Index maxIters = m_maxIters;
	if (maxIters == -1)
		maxIters = m_maxIterationsPerRow * matrixH.rows();
	Scalar* workspace = &m_workspaceVector.coeffRef(0);

	// The matrix m_matT is divided in three parts.
	// Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
	// Rows il,...,iu is the part we are working on (the active window).
	// Rows iu+1,...,end are already brought in triangular form.
	Index iu = m_matT.cols() - 1;
	Index iter = 0;		 // iteration count for current eigenvalue
	Index totalIter = 0; // iteration count for whole matrix
	Scalar exshift(0);	 // sum of exceptional shifts
	Scalar norm = computeNormOfT();
	// sub-diagonal entries smaller than considerAsZero will be treated as zero.
	// We use eps^2 to enable more precision in small eigenvalues.
	Scalar considerAsZero =
		numext::maxi<Scalar>(norm * numext::abs2(NumTraits<Scalar>::epsilon()), (std::numeric_limits<Scalar>::min)());

	if (norm != Scalar(0)) {
		while (iu >= 0) {
			Index il = findSmallSubdiagEntry(iu, considerAsZero);

			// Check for convergence
			if (il == iu) // One root found
			{
				m_matT.coeffRef(iu, iu) = m_matT.coeff(iu, iu) + exshift;
				if (iu > 0)
					m_matT.coeffRef(iu, iu - 1) = Scalar(0);
				iu--;
				iter = 0;
			} else if (il == iu - 1) // Two roots found
			{
				splitOffTwoRows(iu, computeU, exshift);
				iu -= 2;
				iter = 0;
			} else // No convergence yet
			{
				// The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC
				// warning (-O1 -Wall -DNDEBUG )
				Vector3s firstHouseholderVector = Vector3s::Zero(), shiftInfo;
				computeShift(iu, iter, exshift, shiftInfo);
				iter = iter + 1;
				totalIter = totalIter + 1;
				if (totalIter > maxIters)
					break;
				Index im;
				initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
				performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
			}
		}
	}
	if (totalIter <= maxIters)
		m_info = Success;
	else
		m_info = NoConvergence;

	m_isInitialized = true;
	m_matUisUptodate = computeU;
	return *this;
}

/** \internal Computes and returns vector L1 norm of T */
template<typename MatrixType>
inline typename MatrixType::Scalar
RealSchur<MatrixType>::computeNormOfT()
{
	const Index size = m_matT.cols();
	// FIXME to be efficient the following would requires a triangular reduxion code
	// Scalar norm = m_matT.upper().cwiseAbs().sum()
	//               + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
	Scalar norm(0);
	for (Index j = 0; j < size; ++j)
		norm += m_matT.col(j).segment(0, (std::min)(size, j + 2)).cwiseAbs().sum();
	return norm;
}

/** \internal Look for single small sub-diagonal element and returns its index */
template<typename MatrixType>
inline Index
RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, const Scalar& considerAsZero)
{
	using std::abs;
	Index res = iu;
	while (res > 0) {
		Scalar s = abs(m_matT.coeff(res - 1, res - 1)) + abs(m_matT.coeff(res, res));

		s = numext::maxi<Scalar>(s * NumTraits<Scalar>::epsilon(), considerAsZero);

		if (abs(m_matT.coeff(res, res - 1)) <= s)
			break;
		res--;
	}
	return res;
}

/** \internal Update T given that rows iu-1 and iu decouple from the rest. */
template<typename MatrixType>
inline void
RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift)
{
	using std::abs;
	using std::sqrt;
	const Index size = m_matT.cols();

	// The eigenvalues of the 2x2 matrix [a b; c d] are
	// trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
	Scalar p = Scalar(0.5) * (m_matT.coeff(iu - 1, iu - 1) - m_matT.coeff(iu, iu));
	Scalar q = p * p + m_matT.coeff(iu, iu - 1) * m_matT.coeff(iu - 1, iu); // q = tr^2 / 4 - det = discr/4
	m_matT.coeffRef(iu, iu) += exshift;
	m_matT.coeffRef(iu - 1, iu - 1) += exshift;

	if (q >= Scalar(0)) // Two real eigenvalues
	{
		Scalar z = sqrt(abs(q));
		JacobiRotation<Scalar> rot;
		if (p >= Scalar(0))
			rot.makeGivens(p + z, m_matT.coeff(iu, iu - 1));
		else
			rot.makeGivens(p - z, m_matT.coeff(iu, iu - 1));

		m_matT.rightCols(size - iu + 1).applyOnTheLeft(iu - 1, iu, rot.adjoint());
		m_matT.topRows(iu + 1).applyOnTheRight(iu - 1, iu, rot);
		m_matT.coeffRef(iu, iu - 1) = Scalar(0);
		if (computeU)
			m_matU.applyOnTheRight(iu - 1, iu, rot);
	}

	if (iu > 1)
		m_matT.coeffRef(iu - 1, iu - 2) = Scalar(0);
}

/** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
template<typename MatrixType>
inline void
RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
{
	using std::abs;
	using std::sqrt;
	shiftInfo.coeffRef(0) = m_matT.coeff(iu, iu);
	shiftInfo.coeffRef(1) = m_matT.coeff(iu - 1, iu - 1);
	shiftInfo.coeffRef(2) = m_matT.coeff(iu, iu - 1) * m_matT.coeff(iu - 1, iu);

	// Wilkinson's original ad hoc shift
	if (iter == 10) {
		exshift += shiftInfo.coeff(0);
		for (Index i = 0; i <= iu; ++i)
			m_matT.coeffRef(i, i) -= shiftInfo.coeff(0);
		Scalar s = abs(m_matT.coeff(iu, iu - 1)) + abs(m_matT.coeff(iu - 1, iu - 2));
		shiftInfo.coeffRef(0) = Scalar(0.75) * s;
		shiftInfo.coeffRef(1) = Scalar(0.75) * s;
		shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
	}

	// MATLAB's new ad hoc shift
	if (iter == 30) {
		Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
		s = s * s + shiftInfo.coeff(2);
		if (s > Scalar(0)) {
			s = sqrt(s);
			if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
				s = -s;
			s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
			s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
			exshift += s;
			for (Index i = 0; i <= iu; ++i)
				m_matT.coeffRef(i, i) -= s;
			shiftInfo.setConstant(Scalar(0.964));
		}
	}
}

/** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
template<typename MatrixType>
inline void
RealSchur<MatrixType>::initFrancisQRStep(Index il,
										 Index iu,
										 const Vector3s& shiftInfo,
										 Index& im,
										 Vector3s& firstHouseholderVector)
{
	using std::abs;
	Vector3s& v = firstHouseholderVector; // alias to save typing

	for (im = iu - 2; im >= il; --im) {
		const Scalar Tmm = m_matT.coeff(im, im);
		const Scalar r = shiftInfo.coeff(0) - Tmm;
		const Scalar s = shiftInfo.coeff(1) - Tmm;
		v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im + 1, im) + m_matT.coeff(im, im + 1);
		v.coeffRef(1) = m_matT.coeff(im + 1, im + 1) - Tmm - r - s;
		v.coeffRef(2) = m_matT.coeff(im + 2, im + 1);
		if (im == il) {
			break;
		}
		const Scalar lhs = m_matT.coeff(im, im - 1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
		const Scalar rhs =
			v.coeff(0) * (abs(m_matT.coeff(im - 1, im - 1)) + abs(Tmm) + abs(m_matT.coeff(im + 1, im + 1)));
		if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
			break;
	}
}

/** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
template<typename MatrixType>
inline void
RealSchur<MatrixType>::performFrancisQRStep(Index il,
											Index im,
											Index iu,
											bool computeU,
											const Vector3s& firstHouseholderVector,
											Scalar* workspace)
{
	eigen_assert(im >= il);
	eigen_assert(im <= iu - 2);

	const Index size = m_matT.cols();

	for (Index k = im; k <= iu - 2; ++k) {
		bool firstIteration = (k == im);

		Vector3s v;
		if (firstIteration)
			v = firstHouseholderVector;
		else
			v = m_matT.template block<3, 1>(k, k - 1);

		Scalar tau, beta;
		Matrix<Scalar, 2, 1> ess;
		v.makeHouseholder(ess, tau, beta);

		if (beta != Scalar(0)) // if v is not zero
		{
			if (firstIteration && k > il)
				m_matT.coeffRef(k, k - 1) = -m_matT.coeff(k, k - 1);
			else if (!firstIteration)
				m_matT.coeffRef(k, k - 1) = beta;

			// These Householder transformations form the O(n^3) part of the algorithm
			m_matT.block(k, k, 3, size - k).applyHouseholderOnTheLeft(ess, tau, workspace);
			m_matT.block(0, k, (std::min)(iu, k + 3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
			if (computeU)
				m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
		}
	}

	Matrix<Scalar, 2, 1> v = m_matT.template block<2, 1>(iu - 1, iu - 2);
	Scalar tau, beta;
	Matrix<Scalar, 1, 1> ess;
	v.makeHouseholder(ess, tau, beta);

	if (beta != Scalar(0)) // if v is not zero
	{
		m_matT.coeffRef(iu - 1, iu - 2) = beta;
		m_matT.block(iu - 1, iu - 1, 2, size - iu + 1).applyHouseholderOnTheLeft(ess, tau, workspace);
		m_matT.block(0, iu - 1, iu + 1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
		if (computeU)
			m_matU.block(0, iu - 1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
	}

	// clean up pollution due to round-off errors
	for (Index i = im + 2; i <= iu; ++i) {
		m_matT.coeffRef(i, i - 2) = Scalar(0);
		if (i > im + 2)
			m_matT.coeffRef(i, i - 3) = Scalar(0);
	}
}

} // end namespace Eigen

#endif // EIGEN_REAL_SCHUR_H
